# Math Help - Combinatorics

1. ## Combinatorics

Jack has a test with a total of 15 questions. He has to answer 12 of these questions.

In how many ways can he choose which questions to answer if he at least has to choose 3 out of the 5 first questions?

Thank you for taking the time to read my post.

2. Originally Posted by MatteNoob
Jack has a test with a total of 15 questions. He has to answer 12 of these questions.

In how many ways can he choose which questions to answer if he at least has to choose 3 out of the 5 first questions?

Thank you for taking the time to read my post.
I think I've got this, so you would need to evaluate $_5C_3$

$_nC_r=\frac{n!}{r!(n-r)!}$

Thus, $_5C_3=\frac{5!}{3!2!}=\frac{120}{6\times 2}=\color{red}\boxed{10}$

Does this make sense?

--Chris

3. Originally Posted by MatteNoob
Jack has a test with a total of 15 questions. He has to answer 12 of these questions.

In how many ways can he choose which questions to answer if he at least has to choose 3 out of the 5 first questions?

Thank you for taking the time to read my post.
He must answer 3, 4, or 5 out of the first 5 questions.

If he answers 3 out of the first 5 then he must answer 9 out of the last 10; if he answers 4 out of the first 5 then he must answer 8 out of the last 10; and if he answers 5 out of the first 5 then he must answer 7 out of the last 10.

So the total number of possibilities is

$\binom{5}{3} \binom{10}{9} + \binom{5}{4} \binom{10}{8} + \binom{5}{5} \binom{10}{7}$.

4. Thank you both for taking the time to respond to my thread.

I also thought that if he has to choose atleast 3 out of the first five questions he had to do as you said, but that is counter intuitive to me, let me explain why:

$\underbrace{\binom{5}{3} \binom{10}{9}}_{a} + \underbrace{\binom{5}{4} \binom{10}{8}}_{b} + \underbrace{\binom{5}{5} \binom{10}{7}}_{c}$.

Sorry, at further analysis I come to understand that neither a, b or c hold any identical combination of which questions he chose to answer. I appreciate your help.

5. Originally Posted by MatteNoob
Thank you both for taking the time to respond to my thread.

I also thought that if he has to choose atleast 3 out of the first five questions he had to do as you said, but that is counter intuitive to me, let me explain why:

$\underbrace{\binom{5}{3} \binom{10}{9}}_{a} + \underbrace{\binom{5}{4} \binom{10}{8}}_{b} + \underbrace{\binom{5}{5} \binom{10}{7}}_{c}$.

Sorry, at further analysis I come to understand that neither a, b or c hold any identical combination of which questions he chose to answer. I appreciate your help.
MatteNoob,

I am sorry, but I do not understand your question. Why is the answer counter intuitive to you?